Chapter 9: Oscillations
|
|
- Michael Gordon
- 6 years ago
- Views:
Transcription
1 Chaper 9: Ocillaion Now if hi elecron i diplaced fro i equilibriu poiion, a force ha i direcly proporional o he diplaceen reore i like a pendulu o i poiion of re. Pieer Zeean Objecive 1. Decribe he condiion neceary for iple haronic oion.. Wrie down an appropriae expreion for diplaceen of he for Aco(ω) or Ain(ω) decribing he oion. 3. Apply he relaionhip beween frequency, angular frequency, and period. 4. Deerine he oal energy of an objec undergoing iple haronic oion, and kech graph of kineic and poenial energie a funcion of ie or diplaceen. 5. Idenify he inia, axia, and zero of diplaceen, velociy, and acceleraion for an objec undergoing iple haronic oion. 6. Analyze he iple haronic oion of a pring-block ocillaor. 7. Analyze he iple haronic oion of an ideal pendulu. Chaper 9: Ocillaion 1
2 Siple Haronic Moion When an objec i diplaced, i ay be ubjec o a reoring force, reuling in a periodic ocillaing oion. If he diplaceen i direcly proporional o he linear reoring force, he objec undergoe iple haronic oion (SHM). Exaple of linear reoring force cauing iple haronic oion include a a on a pring, he pendulu on a grandfaher clock, a ree lib ocillaing afer you bruh i walking hrough he wood, a child on a wing, even he vibraion of ao in a olid can be odeled a iple haronic oion. Unifor circular oion alo ha ie o iple haronic oion. Conider an objec oving in a horizonal circle of radiu A a conan angular velociy ω a hown in he diagra below. A any given poin in ie, he x-poiion of he objec can be decribed by x=acoθ, and he y-poiion can be decribed by y=ainθ. A x = Aco y = Ain k x=-a x=0 x=a Now iagine a block on a fricionle urface aached o a wall by a pring. If ha block i diplaced fro i equilibriu poiion by an aoun A, and releaed, i iple haronic oion along he fricionle urface will irror he x-oion of he objec oving in unifor circular oion. Siple haronic oion and unifor circular oion are very cloely relaed, a iple haronic oion decribe one dienion of an objec oving in unifor circular oion! Referring back o he chaper on roaional oion, recall ha he angular diplaceen of an objec θ i equal o he angular velociy uliplied by he ie inerval. You can herefore wrie he x-coordinae of he objec in UCM and SHM a θ=ω x = Acoθ x = Aco(ω) In hi equaion, ω i known a he angular frequency, correponding o he nuber of radian per econd for an objec raveling in unifor circular Chaper 9: Ocillaion
3 oion. You can conver fro angular frequency o frequency and period uing he equaion: π ω= πf = T Therefore a general for decribing he oion of an objec undergoing SHM can be wrien a: x() = Aco(ω +φ) The ybol phi (Φ) refer o he phae angle, or aring poin, of a ine or coine curve. Ue he coine curve if he oion ar a axiu apliude. Ue he ine curve if he oion ar a x=0. If he oion being decribed ar oewhere beween axiu apliude and equilibriu (x=0), you ll have o add in a phae angle coponen. A x() = Aco( ) 0 -A 3 A x() = Ain( ) 0 -A 3 If he axiu diplaceen of an objec undergoing SHM i A, hen he axiu peed of he objec i ωa, which you can find by aking he lope of he poiion-ie curve, and i axiu acceleraion i ω A, which you can find by aking he lope of he velociy-ie curve Q: An ocillaing ye i creaed by a releaing an objec fro a axiu diplaceen of 0. eer. The objec ake 60 coplee ocillaion in one inue. Deerine he objec angular frequency A: ω= πf = π(1 Hz) = π rad Chaper 9: Ocillaion 3
4 9.0 Q: Referring o proble 9.01, deerine he objec poiion a ie =10 econd. ω=π rad A=0. = A: x = Aco(ω) x = (0.)co(π) x = (0.)co(π rad 10) = Q: Referring o proble 9.01, deerine he ie when he objec i a poiion x= A: x = Aco(π) co(π) = x A π = x co 1 A x co co 1 A 0. = = = π π rad Noe: if you arrived a an anwer of roughly 9.55 econd, ake ure your calculaor i funcioning in Radian ode inead of Degree ode for hee rigonoeric funcion. Horizonal Spring-Block Ocillaor A popular deonraion vehicle for iple haronic oion i he pringblock ocillaor. The horizonal pring-block ocillaor coni of a block of a iing on a fricionle urface, aached o a verical wall by a pring of pring conan k, a hown in he diagra below. k x=-a x=0 x=a The block i hen diplaced an aoun A fro i equilibriu poiion and allowed o ocillae back and forh. A he block i on a fricionle urface, in he ideal cenario he block would coninue i periodic oion indefiniely. The angular frequency of he block ocillaion can be deerined fro he pring conan and he a. ω = k 4 Chaper 9: Ocillaion
5 Knowing he angular frequency, i quie raighforward hen o calculae he period of ocillaion for he pring-block ocillaor. T S = π ω ω= k T S = π k Noe ha he period of ocillaion depend only on he a of he block and he pring conan. There i no dependency on he agniude of he diplaceen of he block Q: A 5-kg block i aached o a 000 N/ pring a hown and diplaced a diance of 8 c fro i equilibriu poiion before being releaed. 000 N/ 5 kg x=-8 c x=0 x=8 c Deerine he period of ocillaion, he frequency, and he angular frequency for he block. 5kg 9.04 A: T = π = π = S k 000 N 1 1 f = = = 3.18Hz T ω= πf = π(3.18 Hz) = 0 rad 9.05 Q: Rank he following horizonal pring-block ocillaor reing on fricionle urface in er of heir period, fro longe o hore. 500 N/ 10 kg A 50 N/ 7 kg B x=-1 c x=0 x=1 c x=-5 c x=0 x=5 c 000 N/ kg C 1000 N/ 5 kg D x=-8 c x=0 x=8 c x=-15 c x=0 x=15 c 9.05 A: B, A, D, C Chaper 9: Ocillaion 5
6 I alo inereing o look a he energy of he pring-block ocillaor while i undergoing iple haronic oion. Becaue he urface i fricionle, he oal energy of he ye reain conan. However, here i a coninual ranfer of kineic energy ino elaic poenial energy and back. When he block i a i equilibriu poiion, here i no elaic poenial energy ored in he pring, herefore all of he energy of he block i kineic. The block ha achieved i axiu peed. A hi poiion, here i alo no ne force on he block, herefore he block acceleraion i zero. When he block i a i axiu apliude poiion, all of i energy i ored in he pring a elaic poenial energy. For an inan i kineic energy i zero, herefore i velociy i zero. Furher, a hi poiion, he pring exhibi a axiu force on he block, providing he axiu acceleraion. Le ake a look a hi graphically by exaining a pring-block ocillaor a variou poin in i periodic pah. -x x C A B Graph of Relaed Phyical Quaniie X V U K F a 6 Chaper 9: Ocillaion
7 A B C Diplaceen (x) 0 X -X Velociy (v) ax 0 0 Poenial Energy (U) 0 ax ax Kineic Energy (K) ax 0 0 Force (F) 0 -ax ax Acceleraion (a) 0 -ax ax Of coure, hrough he enire ie inerval, he oal echanical energy of he pring-block ocillaor reain conan. In realiy, oe energy i ypically lo o fricion or oher non-conervaive force. Over ie, he apliude of he ocillaion eadily decreae. Thi i known a daping, or daped haronic oion Q: A -kg block i aached o a pring. A force of 0 newon reche he pring o a diplaceen of 50 c. Find: A) he pring conan B) he oal energy C) he peed of he block a he equilibriu poiion D) he peed of he block a x=30 c E) he peed of he block a x=-40 c F) he acceleraion a he equilibriu poiion G) he agniude of he acceleraion a a x=50 c H) he ne force on he block a he equilibriu poiion I) he ne force a x=5 c J) he poiion where he kineic energy i equal o he poenial energy. F 0N 9.06 A: A) k = = = x N 1 1 B) E = U = kx = (40 )(0.5 ) = 5J T S N E (5 J ) C) 1 T U = K = v = E v= = =. S T kg 1 1 D) E = U + K = kx + v E kx = v T S T v = E kx (5 J) (40 N T )(0.3 ) = kg = 1.8 Chaper 9: Ocillaion 7
8 E) v = E kx (5 J) (40 N T )(0.4 ) = kg F) a x=0, F=0, herefore acceleraion = 0 = 1.3 G) = = = kx ( F a kx a = 40 N )(0.5 ) = 10 kg H) A equilibriu, here i no pring diplaceen, o ne force i zero. I) F = kx= ( 40 )(0.5 ) = 10N N E E E T 5J 1 T T J) K = U = kx = x = = = 0.35 S k 40 Noe ha he poin where kineic energy and poenial energy are equal i NOT halfway beween he equilibriu and axiu diplaceen poin. N Spring Cobinaion* I i alo poible o aach ore han one pring o an objec. In hee cae, analye can be iplified coniderably by reaing he cobinaion of pring a a ingle pring wih an equivalen pring conan. For pring in parallel, calculae an equivalen pring conan for he ye by aring wih Hooke Law, recognizing ha diplaceen i he ae for boh pring. k1 k x=-a x=0 x=a F = k x+ k x= ( k + k ) x= k x k = k + k eq eq For pring in erie, you will again calculae an equivalen pring conan for he ye, beginning he analyi by realizing he force on each pring u be he ae according o Newon 3rd Law of Moion. k1 k x=-a x=0 x=a 8 Chaper 9: Ocillaion
9 k F = k x = k x x = k x Nex, recognizing ha he oal diplaceen i equal o he u of he diplaceen of he pring, you can cobine he equaion and olve for an equivalen pring conan. x 1 = k x k F = k eq (x 1 + x ) 1 k F = k F= k eq x k + x x 1 k k x = k eq x +1 k 1 k = k k +1 eq k 1 1 = k eq k 1 k Q: Rank he pring block ocillaor in he diagra below fro highe o lowe in er of: I) equivalen pring conan II) period of ocillaion 10 N/ A 0 N/ 5 N/ 15 N/ 3kg 6kg B -5 c x=0 5 c -6 c x=0 6 c C 15 N/ 15 N/ kg 10 N/ 10 N/ 4kg D -8 c x=0 8 c -7 c x=0 7 c 9.07 A: I) B, D, C, A II) A, C, B, D Verical Spring-Block Ocillaor Spring-block ocillaor can alo be e up verically a hown in he diagra. y=-a k F = ky y=yeq y=a +y g Chaper 9: Ocillaion 9
10 Sar your analyi by drawing a Free Body Diagra for he block, noing ha graviy pull he a down, while he force of he pring provide he upward force. Call down he poiive y-direcion. A i equilibriu poiion, y=y eq. You can hen wrie a Newon nd Law Equaion for he block and olve for y eq. equilibriu F ney = g ky = a y g ky = 0 y eq = g k Once he ye ha eled a equilibriu, you can diplace he a by pulling i oe aoun o eiher +A or lifing i an aoun -A. The new ye can be analyzed a follow: F ney = g k( y eq + A) = g ky eq ka g ky eq =0 F ney = ka Thi i he ae analyi you would do for a horizonal pring ye wih pring conan k diplaced an aoun A fro i equilibriu poiion. Thi ean, in hor, ha o analyze a verical pring ye, all you do i find he new equilibriu poiion of he ye, aking ino accoun he effec of graviy, hen rea i a a ye wih only he pring force o deal wih, ocillaing around he new equilibriu poin. No need o coninue o deal wih he force of graviy! 9.08 Q: A -kg block aached o an unreched pring of pring conan k=00 N/ a hown in he diagra below i releaed fro re. I) Deerine he period of he block ocillaion. II) Wha i he axiu diplaceen of he block fro i equilibriu while undergoing iple haronic oion? kg 00 N/ kg 9.08 A: I) T = π = π = 0.63 S k 00 N II) The graviaional poenial energy a he block aring poin, gδy, u equal he elaic poenial energy ored in he pring a i lowe poin. Ue hi o olve for Δy. U g =U S gδy = 1 kδy Δy = g k If Δy i he oal diplaceen of he block fro i highe poin o i lowe poin, he axiu diplaceen of he block fro i equilibriu poin, A, u be half of Δy. A = Δy = g k = (kg)(9.8 ) = N 30 Chaper 9: Ocillaion
11 9.09 Q: A 5-kg block i aached o a verical pring (k=500 N/). Afer he block coe o re, i i pulled down 3 c and releaed. I) Wha i he period of ocillaion? II) Wha i he axiu diplaceen of he pring fro i iniial unrained poiion? 5kg 9.09 A: I) T = π = π = 0.63 S k 500 N II) Fir deerine he diplaceen of he pring when he block i hanging and a re. g (5 kg)(9.8 ) F = 0 kd = g d = = = 0.1 ney k 500 Then he block i pulled down 3 c, o he axiu diplaceen u be he diplaceen while he block i a re in addiion o he 3 c he block i pulled down. y = = 0.13 ax N Ideal Pendulu Ideal Pendulu provide anoher deonraion vehicle for iple haronic oion. Conider a a aached o a ligh ring ha wing wihou fricion abou he verical equilibriu poiion. A he a ravel along i pah, energy i coninuouly ranferred beween graviaional poenial energy and kineic energy. The reoring force in he cae of he ideal pendulu i provided by graviy. The angular frequency of he ideal pendulu, for all angle of hea, i given by: ω = g l Chaper 9: Ocillaion 31
12 You can hen find he period of he pendulu for all angle of hea: T P = π ω ω= g l T P = π l g Noice ha he period of he pendulu i dependen only upon he lengh of he pendulu and he graviaional field rengh... here i no a dependance! 9.10 Q: A grandfaher clock i deigned uch ha each wing (or half-period) of he pendulu ake one econd. How long i he pendulu in a grandfaher clock? l gt (9.8 )( ) 9.10 A: T = π l = = = 1 P g 4π 4π 9.11 Q: Wha i he period of a grandfaher clock on he oon, where he acceleraion due o graviy on he urface i roughly one-ixh ha of Earh? l A: T = π = π = 4.9 P 1 g (9.8 ) Q: Rank he following pendulu of unifor a deniy fro highe o lowe frequency. A B C D 9.1 A: D, A, B, C 3 Chaper 9: Ocillaion
13 Uilizing he geoeric repreenaion of he pendulu previouly developed in queion 5.7, you can conruc a diagra deailing he energy and force acing on he pendulu a variou poin in i parabolic pah. g g in U g = 0 K ax = g U g = g y U g = gl(1 co ) K = 0 y A he highe poin, a hown on he lef, he a i being pulled back oward i equilibriu poiion by graviy. Specifically, he coponen of graviy along he a pah, ginθ. A he equilibriu poiion, he graviaional poenial energy i a a iniu, and he kineic energy of he a i a a axiu. A he highe poin, a hown on he righ, all he energy i graviaional poenial energy again. Uing he law of conervaion of energy, olving for he axiu velociy of he a a i lowe poiion i quie raighforward. 1 K = U v = gl(1 co θ) v= gl(1 coθ g Furher, fro hi ae graph you can creae a graph of kineic energy, graviaional poenial energy, and oal energy a a poiion of he a along he x-axi. E E o K U g The agniude of he energy below he parabolic line repreen he graviaional poenial energy of he a, while above he line he kineic energy i repreened. The oal echanical energy reain conan hroughou he enire pah of he pendulu. x Chaper 9: Ocillaion 33
14 9.13 Q: The period of an ideal pendulu i T. If he a of he pendulu i ripled while i lengh i quadrupled, wha i he new period of he pendulu? A) 0.5 T B) T C) T D) 4T 9.13 A: C) T 9.14 Q: Which of he following are rue for an ideal pendulu coniing of a a ocillaing back and forh on a ligh ring? (Chooe all ha apply.) A) The kineic energy i alway equal o he poenial energy. B) The axiu force on he pendulu occur when he pendulu ha i axiu kineic energy. C) The acceleraion of he pendulu i zero when he a i a i lowe poin. D) The angular acceleraion of he a reain conan A: C) The acceleraion of he pendulu i zero when he a i a i lowe poin Q: A pendulu of lengh 0 c and a 1 kg i diplaced an angle of 10 degree fro he verical. Wha i he axiu peed of he pendulu? 9.15 A: v= gl(1 co θ) = (9.8 )(0. )(1 co10 ) = Q: A pendulu of lengh 0.5 and a 5 kg i diplaced an angle of 14 degree fro he verical. Wha i he peed of he pendulu when i angle fro he verical i 7 degree? A: U = U + K gl(1 co14 ) = gl(1 co7 ) + v gop g boo boo v= gl(co7 co14 ) = Chaper 9: Ocillaion
15 Te Your Underanding 1. Decribe a lea wo ehod of deerine he pring conan of a pring. Which would you expec o provide ore accurae daa? Explain.. Deign an experien o eaure he acceleraion due o graviy uing a pendulu. Wha equipen would you require? Wha daa would you collec? Wha calculaion would you ake? How would you iniize error? 3. An ideal pendulu i diplaced by an angle hea fro he verical. Fill in he following graph decribing he oion of he pendulu in iilar fahion a hoe decribing he pring-block ocillaor. C A B Graph of Relaed Phyical Quaniie X V U K F a Chaper 9: Ocillaion 35
16 4. A uden eaure he a of a block uing a pring-block ocillaor by eauring he period for he block aached o a variey of pring wih known pring conan. Wha would a graph of he period v. pring conan look like? How could you ue ha graph o deerine he a of he block? 5. How could you double he axiu peed of a 40-kg child on a playground wing? 36 Chaper 9: Ocillaion
Physics 240: Worksheet 16 Name
Phyic 4: Workhee 16 Nae Non-unifor circular oion Each of hee proble involve non-unifor circular oion wih a conan α. (1) Obain each of he equaion of oion for non-unifor circular oion under a conan acceleraion,
More informationLinear Motion, Speed & Velocity
Add Iporan Linear Moion, Speed & Velociy Page: 136 Linear Moion, Speed & Velociy NGSS Sandard: N/A MA Curriculu Fraework (2006): 1.1, 1.2 AP Phyic 1 Learning Objecive: 3.A.1.1, 3.A.1.3 Knowledge/Underanding
More informationHow to Solve System Dynamic s Problems
How o Solve Sye Dynaic Proble A ye dynaic proble involve wo or ore bodie (objec) under he influence of everal exernal force. The objec ay uliaely re, ove wih conan velociy, conan acceleraion or oe cobinaion
More informationPHYSICS 151 Notes for Online Lecture #4
PHYSICS 5 Noe for Online Lecure #4 Acceleraion The ga pedal in a car i alo called an acceleraor becaue preing i allow you o change your elociy. Acceleraion i how fa he elociy change. So if you ar fro re
More informationDiscussion Session 2 Constant Acceleration/Relative Motion Week 03
PHYS 100 Dicuion Seion Conan Acceleraion/Relaive Moion Week 03 The Plan Today you will work wih your group explore he idea of reference frame (i.e. relaive moion) and moion wih conan acceleraion. You ll
More informationEnergy Problems 9/3/2009. W F d mgh m s 196J 200J. Understanding. Understanding. Understanding. W F d. sin 30
9/3/009 nderanding Energy Proble Copare he work done on an objec o a.0 kg a) In liing an objec 0.0 b) Puhing i up a rap inclined a 30 0 o he ae inal heigh 30 0 puhing 0.0 liing nderanding Copare he work
More informationThus the force is proportional but opposite to the displacement away from equilibrium.
Chaper 3 : Siple Haronic Moion Hooe s law saes ha he force (F) eered by an ideal spring is proporional o is elongaion l F= l where is he spring consan. Consider a ass hanging on a he spring. In equilibriu
More informationChapter 8 Torque and Angular Momentum
Chaper 8 Torque and Angular Moenu Reiew of Chaper 5 We had a able coparing paraeer fro linear and roaional oion. Today we fill in he able. Here i i Decripion Linear Roaional poiion diplaceen Rae of change
More informationSHM SHM. T is the period or time it takes to complete 1 cycle. T = = 2π. f is the frequency or the number of cycles completed per unit time.
SHM A ω = k d x x = Acos ( ω +) dx v = = ω Asin( ω + ) vax = ± ωa dv a = = ω Acos + k + x Apliude ( ω ) = 0 a ax = ± ω A SHM x = Acos is he period or ie i akes o coplee cycle. ω = π ( ω +) π = = π ω k
More information13.1 Accelerating Objects
13.1 Acceleraing Objec A you learned in Chaper 12, when you are ravelling a a conan peed in a raigh line, you have uniform moion. However, mo objec do no ravel a conan peed in a raigh line o hey do no
More informationx y θ = 31.8 = 48.0 N. a 3.00 m/s
4.5.IDENTIY: Vecor addiion. SET UP: Use a coordinae sse where he dog A. The forces are skeched in igure 4.5. EXECUTE: + -ais is in he direcion of, A he force applied b =+ 70 N, = 0 A B B A = cos60.0 =
More informationIntroduction to Numerical Analysis. In this lesson you will be taken through a pair of techniques that will be used to solve the equations of.
Inroducion o Nuerical Analysis oion In his lesson you will be aen hrough a pair of echniques ha will be used o solve he equaions of and v dx d a F d for siuaions in which F is well nown, and he iniial
More informationAngular Motion, Speed and Velocity
Add Imporan Angular Moion, Speed and Velociy Page: 163 Noe/Cue Here Angular Moion, Speed and Velociy NGSS Sandard: N/A MA Curriculum Framework (006): 1.1, 1. AP Phyic 1 Learning Objecive: 3.A.1.1, 3.A.1.3
More informationIB Physics Kinematics Worksheet
IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?
More informationLecture 23 Damped Motion
Differenial Equaions (MTH40) Lecure Daped Moion In he previous lecure, we discussed he free haronic oion ha assues no rearding forces acing on he oving ass. However No rearding forces acing on he oving
More informationChapter 15 Oscillatory Motion I
Chaper 15 Oscillaory Moion I Level : AP Physics Insrucor : Kim Inroducion A very special kind of moion occurs when he force acing on a body is proporional o he displacemen of he body from some equilibrium
More information1. VELOCITY AND ACCELERATION
1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under
More informationPROBLEMS ON RECTILINEAR MOTION
PROBLEMS ON RECTILINEAR MOTION PROBLEM 1. The elociy of a paricle which oe along he -ai i gien by 5 (/). Ealuae he diplaceen elociy and acceleraion a when = 4. The paricle i a he origin = when =. (/) /
More informationCHAPTER 7: SECOND-ORDER CIRCUITS
EEE5: CI RCUI T THEORY CHAPTER 7: SECOND-ORDER CIRCUITS 7. Inroducion Thi chaper conider circui wih wo orage elemen. Known a econd-order circui becaue heir repone are decribed by differenial equaion ha
More informationv 1 =4 m/s v 2 =0 m 1 =0.5kg m 2 Momentum F (N) t (s) v 0y v x
Moenu Do our work on a earae hee of aer or noebook. or each roble, draw clearl labeled diagra howing he ae and elociie for each objec before and afer he colliion. Don forge abou direcion oenu, eloci and
More informationMotion In One Dimension. Graphing Constant Speed
Moion In One Dimenion PLATO AND ARISTOTLE GALILEO GALILEI LEANING TOWER OF PISA Graphing Conan Speed Diance v. Time for Toy Car (0-5 ec.) be-fi line (from TI calculaor) d = 207.7 12.6 Diance (cm) 1000
More informationHomework 2 Solutions
Mah 308 Differenial Equaions Fall 2002 & 2. See he las page. Hoework 2 Soluions 3a). Newon s secon law of oion says ha a = F, an we know a =, so we have = F. One par of he force is graviy, g. However,
More informationViscous Damping Summary Sheet No Damping Case: Damped behaviour depends on the relative size of ω o and b/2m 3 Cases: 1.
Viscous Daping: && + & + ω Viscous Daping Suary Shee No Daping Case: & + ω solve A ( ω + α ) Daped ehaviour depends on he relaive size of ω o and / 3 Cases:. Criical Daping Wee 5 Lecure solve sae BC s
More information4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More informationNotes on MRI, Part II
BME 483 MRI Noe : page 1 Noe on MRI, Par II Signal Recepion in MRI The ignal ha we deec in MRI i a volage induced in an RF coil by change in agneic flu fro he preceing agneizaion in he objec. One epreion
More information5.2 GRAPHICAL VELOCITY ANALYSIS Polygon Method
ME 352 GRHICL VELCITY NLYSIS 52 GRHICL VELCITY NLYSIS olygon Mehod Velociy analyi form he hear of kinemaic and dynamic of mechanical yem Velociy analyi i uually performed following a poiion analyi; ie,
More informationElastic and Inelastic Collisions
laic and Inelaic Colliion In an LASTIC colliion, energy i conered (Kbefore = Kafer or Ki = Kf. In an INLASTIC colliion, energy i NOT conered. (Ki > Kf. aple: A kg block which i liding a 0 / acro a fricionle
More informationSuggested Practice Problems (set #2) for the Physics Placement Test
Deparmen of Physics College of Ars and Sciences American Universiy of Sharjah (AUS) Fall 014 Suggesed Pracice Problems (se #) for he Physics Placemen Tes This documen conains 5 suggesed problems ha are
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
More informationMon Apr 2: Laplace transform and initial value problems like we studied in Chapter 5
Mah 225-4 Week 2 April 2-6 coninue.-.3; alo cover par of.4-.5, EP 7.6 Mon Apr 2:.-.3 Laplace ranform and iniial value problem like we udied in Chaper 5 Announcemen: Warm-up Exercie: Recall, The Laplace
More informationCurvature. Institute of Lifelong Learning, University of Delhi pg. 1
Dicipline Coure-I Semeer-I Paper: Calculu-I Leon: Leon Developer: Chaianya Kumar College/Deparmen: Deparmen of Mahemaic, Delhi College of r and Commerce, Univeriy of Delhi Iniue of Lifelong Learning, Univeriy
More information2. VECTORS. R Vectors are denoted by bold-face characters such as R, V, etc. The magnitude of a vector, such as R, is denoted as R, R, V
ME 352 VETS 2. VETS Vecor algebra form he mahemaical foundaion for kinemaic and dnamic. Geomer of moion i a he hear of boh he kinemaic and dnamic of mechanical em. Vecor anali i he imehonored ool for decribing
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationProblem Set If all directed edges in a network have distinct capacities, then there is a unique maximum flow.
CSE 202: Deign and Analyi of Algorihm Winer 2013 Problem Se 3 Inrucor: Kamalika Chaudhuri Due on: Tue. Feb 26, 2013 Inrucion For your proof, you may ue any lower bound, algorihm or daa rucure from he ex
More informationRandomized Perfect Bipartite Matching
Inenive Algorihm Lecure 24 Randomized Perfec Biparie Maching Lecurer: Daniel A. Spielman April 9, 208 24. Inroducion We explain a randomized algorihm by Ahih Goel, Michael Kapralov and Sanjeev Khanna for
More informationNon-uniform circular motion *
OpenSax-CNX module: m14020 1 Non-uniform circular moion * Sunil Kumar Singh This work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License 2.0 Wha do we mean by non-uniform
More informationChapter 1 Rotational dynamics 1.1 Angular acceleration
Chaper Roaional dynamics. Angular acceleraion Learning objecives: Wha do we mean by angular acceleraion? How can we calculae he angular acceleraion of a roaing objec when i speeds up or slows down? How
More information0 time. 2 Which graph represents the motion of a car that is travelling along a straight road with a uniformly increasing speed?
1 1 The graph relaes o he moion of a falling body. y Which is a correc descripion of he graph? y is disance and air resisance is negligible y is disance and air resisance is no negligible y is speed and
More informationWhen analyzing an object s motion there are two factors to consider when attempting to bring it to rest. 1. The object s mass 2. The object s velocity
SPH4U Momenum LoRuo Momenum i an exenion of Newon nd law. When analyzing an ojec moion here are wo facor o conider when aeming o ring i o re.. The ojec ma. The ojec velociy The greaer an ojec ma, he more
More informationDisplacement ( x) x x x
Kinemaics Kinemaics is he branch of mechanics ha describes he moion of objecs wihou necessarily discussing wha causes he moion. 1-Dimensional Kinemaics (or 1- Dimensional moion) refers o moion in a sraigh
More informationPhysics 20 Lesson 28 Simple Harmonic Motion Dynamics & Energy
Phyic 0 Leon 8 Siple Haronic Motion Dynaic & Energy Now that we hae learned about work and the Law of Coneration of Energy, we are able to look at how thee can be applied to the ae phenoena. In general,
More informationSection 14 Forces in Circular Motion
Secion 14 orces in Circular Moion Ouline 1 Unifor Circular Moion Non-unifor Circular Moion Phsics 04A Class Noes Wh do objecs do wha he do? The answer we have been invesigaing is forces If forces can eplain
More informationEF 151 Exam #2 - Spring, 2014 Page 1 of 6
EF 5 Exam # - Spring, 04 Page of 6 Name: Secion: Inrucion: Pu your name and ecion on he exam. Do no open he e unil you are old o do o. Wrie your final anwer in he box proided If you finih wih le han 5
More informationOscillations. Periodic Motion. Sinusoidal Motion. PHY oscillations - J. Hedberg
Oscillaions PHY 207 - oscillaions - J. Hedberg - 2017 1. Periodic Moion 2. Sinusoidal Moion 3. How do we ge his kind of moion? 4. Posiion - Velociy - cceleraion 5. spring wih vecors 6. he reference circle
More informationA man pushes a 500 kg block along the x axis by a constant force. Find the power required to maintain a speed of 5.00 m/s.
Coordinaor: Dr. F. hiari Wednesday, July 16, 2014 Page: 1 Q1. The uniform solid block in Figure 1 has mass 0.172 kg and edge lenghs a = 3.5 cm, b = 8.4 cm, and c = 1.4 cm. Calculae is roaional ineria abou
More information~v = x. ^x + ^y + ^x + ~a = vx. v = v 0 + at. ~v P=A = ~v P=B + ~v B=A. f k = k. W tot =KE. P av =W=t. W grav = mgy 1, mgy 2 = mgh =,U grav
PHYSICS 5A FALL 2001 FINAL EXAM v = x a = v x = 1 2 a2 + v 0 + x 0 v 2 = v 2 0 +2a(x, x 0) a = v2 r ~v = x ~a = vx v = v 0 + a y z ^x + ^y + ^z ^x + vy x, x 0 = 1 2 (v 0 + v) ~v P=A = ~v P=B + ~v B=A ^y
More informationWeek 1 Lecture 2 Problems 2, 5. What if something oscillates with no obvious spring? What is ω? (problem set problem)
Week 1 Lecure Problems, 5 Wha if somehing oscillaes wih no obvious spring? Wha is ω? (problem se problem) Sar wih Try and ge o SHM form E. Full beer can in lake, oscillaing F = m & = ge rearrange: F =
More informationAlgorithmic Discrete Mathematics 6. Exercise Sheet
Algorihmic Dicree Mahemaic. Exercie Shee Deparmen of Mahemaic SS 0 PD Dr. Ulf Lorenz 7. and 8. Juni 0 Dipl.-Mah. David Meffer Verion of June, 0 Groupwork Exercie G (Heap-Sor) Ue Heap-Sor wih a min-heap
More informationRectilinear Kinematics
Recilinear Kinemaic Coninuou Moion Sir Iaac Newon Leonard Euler Oeriew Kinemaic Coninuou Moion Erraic Moion Michael Schumacher. 7-ime Formula 1 World Champion Kinemaic The objecie of kinemaic i o characerize
More informationAP Calculus BC Chapter 10 Part 1 AP Exam Problems
AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a
More informationExam #2 PHYSICS 211 Monday July 6 th, 2009 Please write down your name also on the back page of this exam
Exa #2 PHYSICS 211 Monday July 6 h, 29 NME Please wrie down your nae also on he back pae of his exa 1. The fiure ives how he force varies as a funcion of he posiion. Such force is acin on a paricle, which
More informationChapter 11 VIBRATORY MOTION
Ch. 11--Vibraory Moion Chaper 11 VIBRATORY MOTION Noe: There are wo areas of ineres when discussing oscillaory moion: he mahemaical characerizaion of vibraing srucures ha generae waves and he ineracion
More informationLab #2: Kinematics in 1-Dimension
Reading Assignmen: Chaper 2, Secions 2-1 hrough 2-8 Lab #2: Kinemaics in 1-Dimension Inroducion: The sudy of moion is broken ino wo main areas of sudy kinemaics and dynamics. Kinemaics is he descripion
More information1 Motivation and Basic Definitions
CSCE : Deign and Analyi of Algorihm Noe on Max Flow Fall 20 (Baed on he preenaion in Chaper 26 of Inroducion o Algorihm, 3rd Ed. by Cormen, Leieron, Rive and Sein.) Moivaion and Baic Definiion Conider
More information6.302 Feedback Systems Recitation : Phase-locked Loops Prof. Joel L. Dawson
6.32 Feedback Syem Phae-locked loop are a foundaional building block for analog circui deign, paricularly for communicaion circui. They provide a good example yem for hi cla becaue hey are an excellen
More information2. What is the displacement of the bug between t = 0.00 s and t = 20.0 s? A) cm B) 39.9 cm C) cm D) 16.1 cm E) +16.
1. For which one of he following siuaions will he pah lengh equal he magniude of he displacemen? A) A jogger is running around a circular pah. B) A ball is rolling down an inclined plane. C) A rain ravels
More informationIn this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper The moion of projeciles In his chaper he model of free moion
More informationPhysics 20 Lesson 5 Graphical Analysis Acceleration
Physics 2 Lesson 5 Graphical Analysis Acceleraion I. Insananeous Velociy From our previous work wih consan speed and consan velociy, we know ha he slope of a posiion-ime graph is equal o he velociy of
More informationEF 151 Exam #1, Spring, 2009 Page 1 of 6
EF 5 Exam #, Spring, 009 Page of 6 Name: Guideline: Aume 3 ignifican figure for all given number unle oherwie aed Show all of your work no work, no credi Wrie your final anwer in he box provided Include
More information6 December 2013 H. T. Hoang - www4.hcmut.edu.vn/~hthoang/ 1
Lecure Noe Fundamenal of Conrol Syem Inrucor: Aoc. Prof. Dr. Huynh Thai Hoang Deparmen of Auomaic Conrol Faculy of Elecrical & Elecronic Engineering Ho Chi Minh Ciy Univeriy of Technology Email: hhoang@hcmu.edu.vn
More information2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance
Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion
More informationMotion along a Straight Line
chaper 2 Moion along a Sraigh Line verage speed and average velociy (Secion 2.2) 1. Velociy versus speed Cone in he ebook: fer Eample 2. Insananeous velociy and insananeous acceleraion (Secions 2.3, 2.4)
More informationMOMENTUM CONSERVATION LAW
1 AAST/AEDT AP PHYSICS B: Impulse and Momenum Le us run an experimen: The ball is moving wih a velociy of V o and a force of F is applied on i for he ime inerval of. As he resul he ball s velociy changes
More informationChapter 3 Kinematics in Two Dimensions
Chaper 3 KINEMATICS IN TWO DIMENSIONS PREVIEW Two-dimensional moion includes objecs which are moing in wo direcions a he same ime, such as a projecile, which has boh horizonal and erical moion. These wo
More informationIn a shop window an illuminated spot on a display oscillates between positions W and Z with simple harmonic motion.
Quesions 1 and 2 refer o he informaion below. In a shop window an illuminaed spo on a display oscillaes beween posiions W and Z wih simple harmonic moion. The diagram shows he display wih a scale added.
More informationIntroduction to Congestion Games
Algorihmic Game Theory, Summer 2017 Inroducion o Congeion Game Lecure 1 (5 page) Inrucor: Thoma Keelheim In hi lecure, we ge o know congeion game, which will be our running example for many concep in game
More informationPhysics 101 Fall 2006: Exam #1- PROBLEM #1
Physics 101 Fall 2006: Exam #1- PROBLEM #1 1. Problem 1. (+20 ps) (a) (+10 ps) i. +5 ps graph for x of he rain vs. ime. The graph needs o be parabolic and concave upward. ii. +3 ps graph for x of he person
More informationConstant Acceleration
Objecive Consan Acceleraion To deermine he acceleraion of objecs moving along a sraigh line wih consan acceleraion. Inroducion The posiion y of a paricle moving along a sraigh line wih a consan acceleraion
More informationWORK, ENERGY AND POWER NCERT
Exemplar Problems Physics Chaper Six WORK, ENERGY AND POWER MCQ I 6.1 An elecron and a proon are moving under he influence of muual forces. In calculaing he change in he kineic energy of he sysem during
More informationQ.1 Define work and its unit?
CHP # 6 ORK AND ENERGY Q.1 Define work and is uni? A. ORK I can be define as when we applied a force on a body and he body covers a disance in he direcion of force, hen we say ha work is done. I is a scalar
More informationTP A.14 The effects of cut angle, speed, and spin on object ball throw
echnical proof echnical proof TP A.14 The effecs of cu angle, speed, and spin on objec ball hrow supporing: The Illusraed Principles of Pool and illiards hp://billiards.colosae.edu by Daid G. Alciaore,
More informationSPH3U: Projectiles. Recorder: Manager: Speaker:
SPH3U: Projeciles Now i s ime o use our new skills o analyze he moion of a golf ball ha was ossed hrough he air. Le s find ou wha is special abou he moion of a projecile. Recorder: Manager: Speaker: 0
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan
Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure
More informationKinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.
Kinemaics Vocabulary Kinemaics and One Dimensional Moion 8.1 WD1 Kinema means movemen Mahemaical descripion of moion Posiion Time Inerval Displacemen Velociy; absolue value: speed Acceleraion Averages
More informationTP B.2 Rolling resistance, spin resistance, and "ball turn"
echnical proof TP B. olling reiance, pin reiance, and "ball urn" upporing: The Illuraed Principle of Pool and Billiard hp://billiard.coloae.edu by Daid G. Alciaore, PhD, PE ("Dr. Dae") echnical proof originally
More informationwhere the coordinate X (t) describes the system motion. X has its origin at the system static equilibrium position (SEP).
Appendix A: Conservaion of Mechanical Energy = Conservaion of Linear Momenum Consider he moion of a nd order mechanical sysem comprised of he fundamenal mechanical elemens: ineria or mass (M), siffness
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationBayesian Designs for Michaelis-Menten kinetics
Bayeian Deign for ichaeli-enen kineic John ahew and Gilly Allcock Deparen of Saiic Univeriy of Newcale upon Tyne.n..ahew@ncl.ac.uk Reference ec. on hp://www.a.ncl.ac.uk/~nn/alk/ile.h Enzyology any biocheical
More informationChapter 12: Velocity, acceleration, and forces
To Feel a Force Chaper Spring, Chaper : A. Saes of moion For moion on or near he surface of he earh, i is naural o measure moion wih respec o objecs fixed o he earh. The 4 hr. roaion of he earh has a measurable
More informationSample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems
Sample Final Exam Covering Chaper 9 (final04) Sample Final Exam (final03) Covering Chaper 9 of Fundamenal of Signal & Syem Problem (0 mar) Conider he caual opamp circui iniially a re depiced below. I LI
More informationParametrics and Vectors (BC Only)
Paramerics and Vecors (BC Only) The following relaionships should be learned and memorized. The paricle s posiion vecor is r() x(), y(). The velociy vecor is v(),. The speed is he magniude of he velociy
More informationDecimal moved after first digit = 4.6 x Decimal moves five places left SCIENTIFIC > POSITIONAL. a) g) 5.31 x b) 0.
PHYSICS 20 UNIT 1 SCIENCE MATH WORKSHEET NAME: A. Sandard Noaion Very large and very small numbers are easily wrien using scienific (or sandard) noaion, raher han decimal (or posiional) noaion. Sandard
More informationSolution: b All the terms must have the dimension of acceleration. We see that, indeed, each term has the units of acceleration
PHYS 54 Tes Pracice Soluions Spring 8 Q: [4] Knowing ha in he ne epression a is acceleraion, v is speed, is posiion and is ime, from a dimensional v poin of view, he equaion a is a) incorrec b) correc
More informationWelcome Back to Physics 215!
Welcome Back o Physics 215! (General Physics I) Thurs. Jan 19 h, 2017 Lecure01-2 1 Las ime: Syllabus Unis and dimensional analysis Today: Displacemen, velociy, acceleraion graphs Nex ime: More acceleraion
More informationMEI Mechanics 1 General motion. Section 1: Using calculus
Soluions o Exercise MEI Mechanics General moion Secion : Using calculus. s 4 v a 6 4 4 When =, v 4 a 6 4 6. (i) When = 0, s = -, so he iniial displacemen = - m. s v 4 When = 0, v = so he iniial velociy
More informationIntroduction to SLE Lecture Notes
Inroducion o SLE Lecure Noe May 13, 16 - The goal of hi ecion i o find a ufficien condiion of λ for he hull K o be generaed by a imple cure. I urn ou if λ 1 < 4 hen K i generaed by a imple curve. We will
More informationPhysics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.
Physics 180A Fall 2008 Tes 1-120 poins Name Provide he bes answer o he following quesions and problems. Wach your sig figs. 1) The number of meaningful digis in a number is called he number of. When numbers
More informationEE202 Circuit Theory II
EE202 Circui Theory II 2017-2018, Spring Dr. Yılmaz KALKAN I. Inroducion & eview of Fir Order Circui (Chaper 7 of Nilon - 3 Hr. Inroducion, C and L Circui, Naural and Sep epone of Serie and Parallel L/C
More informationdp dt For the time interval t, approximately, we can write,
PHYSICS OCUS 58 So far we hae deal only wih syses in which he oal ass of he syse, sys, reained consan wih ie. Now, we will consider syses in which ass eners or leaes he syse while we are obsering i. The
More informationTo become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship
Laplace Tranform (Lin & DeCarlo: Ch 3) ENSC30 Elecric Circui II The Laplace ranform i an inegral ranformaion. I ranform: f ( ) F( ) ime variable complex variable From Euler > Lagrange > Laplace. Hence,
More informationChapter 6. Laplace Transforms
6- Chaper 6. Laplace Tranform 6.4 Shor Impule. Dirac Dela Funcion. Parial Fracion 6.5 Convoluion. Inegral Equaion 6.6 Differeniaion and Inegraion of Tranform 6.7 Syem of ODE 6.4 Shor Impule. Dirac Dela
More informationThe Residual Graph. 12 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm
Augmening Pah Algorihm Greedy-algorihm: ar wih f (e) = everywhere find an - pah wih f (e) < c(e) on every edge augmen flow along he pah repea a long a poible The Reidual Graph From he graph G = (V, E,
More informationPHYSICS 149: Lecture 9
PHYSICS 149: Lecure 9 Chaper 3 3.2 Velociy and Acceleraion 3.3 Newon s Second Law of Moion 3.4 Applying Newon s Second Law 3.5 Relaive Velociy Lecure 9 Purdue Universiy, Physics 149 1 Velociy (m/s) The
More information6.8 Laplace Transform: General Formulas
48 HAP. 6 Laplace Tranform 6.8 Laplace Tranform: General Formula Formula Name, ommen Sec. F() l{ f ()} e f () d f () l {F()} Definiion of Tranform Invere Tranform 6. l{af () bg()} al{f ()} bl{g()} Lineariy
More informationAdmin MAX FLOW APPLICATIONS. Flow graph/networks. Flow constraints 4/30/13. CS lunch today Grading. in-flow = out-flow for every vertex (except s, t)
/0/ dmin lunch oday rading MX LOW PPLIION 0, pring avid Kauchak low graph/nework low nework direced, weighed graph (V, ) poiive edge weigh indicaing he capaciy (generally, aume ineger) conain a ingle ource
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual
More informationPosition, Velocity, and Acceleration
rev 06/2017 Posiion, Velociy, and Acceleraion Equipmen Qy Equipmen Par Number 1 Dynamic Track ME-9493 1 Car ME-9454 1 Fan Accessory ME-9491 1 Moion Sensor II CI-6742A 1 Track Barrier Purpose The purpose
More information